# How do you find the derivative of #y=tan(3x)# ?

Since there is a function inside a function (a "composite" function), you could use the chain rule to accomplish this. It is as follows:

The derivative of a composite function F(x) is as follows:

Alternatively put:

= the derivative of the inner function multiplied by the derivative of the outer function when the inner function is left unaltered.

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To find the derivative of (y = \tan(3x)), you'll use the chain rule, which is a fundamental technique in calculus for differentiating composite functions. The chain rule states that if you have a composite function (f(g(x))), then its derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. The derivative of (\tan(x)) is (\sec^2(x)), and the derivative of (3x) with respect to (x) is 3.

So, applying the chain rule to (y = \tan(3x)), you get:

[ \frac{dy}{dx} = \sec^2(3x) \times \frac{d}{dx}(3x) ]

[ \frac{dy}{dx} = 3\sec^2(3x) ]

Therefore, the derivative of (y = \tan(3x)) with respect to (x) is (3\sec^2(3x)).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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